Find the rate of change of total revenue, cost, and profit with respect to time. Assume that $R(x)$ and $C(x)$ are in dollars. $R(x)=55 x-0.5 x^{2}, C(x)=2 x+20$, when $x=25$ and $d x / d t=20$ units per day The rate of change of total revenue is $\$ \square$ per day.

Step 1 ：Given the revenue function \(R(x) = 55x - 0.5x^2\), the cost function \(C(x) = 2x + 20\), and \(dx/dt = 20\) units per day when \(x = 25\).

Step 2 ：We need to find the rate of change of total revenue with respect to time. This can be found by taking the derivative of the revenue function with respect to x, and then multiplying by the rate of change of x with respect to time.

Step 3 ：First, we find the derivative of the revenue function with respect to x: \(dR/dx = 55 - x\).

Step 4 ：Then, we substitute \(x = 25\) into \(dR/dx\) to get \(dR/dx = 55 - 25 = 30\).

Step 5 ：Finally, we multiply \(dR/dx\) by \(dx/dt\) to get the rate of change of total revenue with respect to time: \(dR/dt = dR/dx * dx/dt = 30 * 20 = 600\) dollars per day.

Step 6 ：Final Answer: The rate of change of total revenue with respect to time is \(\boxed{600}\) dollars per day.